Moment of Inertia

Moment of Inertia Defined  The moment of inertia measures the resistance to a change in rotation. Change in rotation from torqueChange in rotation from torque Moment of inertia I = mr 2 for a single massMoment of inertia I = mr 2 for a single mass  The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.

Two Spheres  A spun baton has a moment of inertia due to each separate mass. I = mr 2 + mr 2 = 2mr 2  If it spins around one end, only the far mass counts. I = m(2r) 2 = 4mr 2 m r m

Mass at a Radius  Extended objects can be treated as a sum of small masses.  A straight rod ( M ) is a set of identical masses  m.  The total moment of inertia is  Each mass element contributes  The sum becomes an integral axis length L distance r to r+  r

Rigid Body Rotation  The moments of inertia for many shapes can found by integration. Ring or hollow cylinder: I = MR 2Ring or hollow cylinder: I = MR 2 Solid cylinder: I = (1/2) MR 2Solid cylinder: I = (1/2) MR 2 Hollow sphere: I = (2/3) MR 2Hollow sphere: I = (2/3) MR 2 Solid sphere: I = (2/5) MR 2Solid sphere: I = (2/5) MR 2

Point and Ring  The point mass, ring and hollow cylinder all have the same moment of inertia. I = MR 2I = MR 2  All the mass is equally far away from the axis.  The rod and rectangular plate also have the same moment of inertia. I = (1/3) MR 2  The distribution of mass from the axis is the same. R M M R axis length R M M

Playground Ride  A child of 180 N sits at the edge of a merry-go-round with radius 2.0 m and mass 160 kg.  What is the moment of inertia, including the child?  Assume the merry-go-round is a disk. I d = (1/2)Mr 2 = 320 kg m 2  Treat the child as a point mass. W = mg, m = W/g = 18 kg. I c = mr 2 = 72 kg m 2  The total moment of inertia is the sum. I = I d + I c = 390 kg m 2 m M r

Parallel Axis Theorem  Some objects don’t rotate about the axis at the center of mass.  The moment of inertia depends on the distance between axes.  The moment of inertia for a rod about its center of mass: axis M h = R/2

Perpendicular Axis Theorem  For flat objects the rotational moment of inertia of the axes in the plane is related to the moment of inertia perpendicular to the plane. M I x = (1/12) Mb 2 I y = (1/12) Ma 2 a b I z = (1/12) M(a 2 + b 2 )

Spinning Coin  What is the moment of inertia of a coin of mass M and radius R spinning on one edge?  The moment of inertia of a spinning disk perpendicular to the plane is known. I d = (1/2) MR 2  The disk has two equal axes in the plane.  The perpendicular axis theorem links these. I d = I e + I e = (1/2) MR 2 I e = (1/4) MR 2 M R M R next IdId IeIe